Abstract

Starting with a trustworthy theory T, Galvan (1992) suggests to read offš, from the usual hierarchy of theories determined by consistency strength, a finer-grained hierarchy in which theories higher up are capable of ‘explaining’, though not fully justifying, our commitment to theories lower down. One way to ascend Galvan’s ‘hierarchy of explanation’ is to formalize soundness proofs: to this extent it often suffices to assume a full theory of truth for the theory T whose soundness is at stake. In this paper, we investigate the possibility of an extension of this method. Our ultimate goal will be to extend T not only with truth axioms, but with a combination of axioms for predicates for truth and necessity. We first consider two alternative strategies for providing possible worlds semantics for necessity as a predicate, one based on classical logic, the other on a supervaluationist interpretation of necessity. We will then formulate a deductive system of truth and necessity in classical logic that is sound with respect to the given (nonclassical) semantics.